What is the remainder when $1! + 2! + 3! + \dots + 200!$ is divided by $14$?

$A$ man $X$ has $7$ friends,$4$ of them are ladies and $3$ are men. His wife $Y$ also has $7$ friends,$3$ of them are ladies and $4$ are men. Assume $X$ and $Y$ have no common friends. The total number of ways in which $X$ and $Y$ together can throw a party inviting $3$ ladies and $3$ men,such that $3$ friends of each of $X$ and $Y$ are invited,is:

$A$ question paper is divided into two parts $A$ and $B$,and each part contains $5$ questions. The number of ways in which a candidate can answer $6$ questions,selecting at least two questions from each part,is:

The total number of three-digit numbers,where exactly one digit is repeated two times,is

The numbers $1, 2, 3, \ldots, n$ are arranged in a random order. The probability that the digits $1, 2, 3, \ldots, k$ appear as a block in that order is:

Let $A = \{(a, b, c) : a, b, c \text{ are non-negative integers and } a + b + 2c = 22\}$. Then $n(A)$ is equal to: